Let • (a) be a function such that A$(x) = U. zo .:: $ (a +1)-0(a)= Ua, $ (a + 2) - (a + 1) = Ua+s, $ (x) - (x - 1) = Uz-1) :: $ (w) - € (a) = Ua + W44p...... + Wx-1 = Eu, in (3). Hence retaining for Eu, the definition of (4) we should write (3) thus: *Euz - Eva = Ua + Wa41 ...... + Uze=2** + U2_2 . . . . . . .....(5). Again suppose Eu, to be defined by (3) and be equal to $ (x), and let the Eu, of (4) be given generally by 0 (0) + W.7 then u. = A {$ (x) + wr} = A() + Aw.= U, +Awx; .. Aw. = 0, or w. does not change when x is increased by unity; hence it remains constant while x takes all the series of values which it is permitted to take in any problem in Finite Differences. Since then we will remain unchanged, so far as we shall have to do with it, we shall denote it by C and regard it as a constant, and examine its true nature later on. (Art. 4, Ch. 11.) Hence regarding Euz as defined by (3) we should write (4) thus: + * Were it not that in so fundamental a theorem it is advisable to use only such methods as are beyond all suspicion as to their rigour, we might have arrived more easily at the same result symbolically, thus: Wa't Watit + Uze1={1+E+E? + ... +E*-a-1}U, E--- -1 Ua=(E*-a – 1) A-14a=(Ez-- 1) Eua, from (4)...(7), E-1 = Σu. - Σμα............... which agrees with (5). But the method in the text is preferable, since the steps in (7) and (8) presuppose a rigorous examination into the nature of the symbols A-1 and 3 before we can state the arithmetical equivalence of the quantities with which we are dealing, i.e. some such investigation as that in the text. ..(8), We shall not dwell farther on this point, since the difference between the Euz of (3) and that of (4) is precisely analogous to that between the definite integral °°(a)dx , and the indefinite integral Jø () do, and the precautions necessary to be taken in using them are identical with those to which we are accustomed in the Integral Calculus. In fact we adopt a notation for definite Finite Integrals strikingly similar to that for Definite Integrals in the Infinitesimal Calculus, writing the Euz of (3) in the form ge=2-1 Integrable Forms. 2. As in Integral Calculus, we shall be able to obtain finite expressions for the integrals of but few forms, and must be content to express the integrals of others in the form of infinite series. Of such integrable forms the following are the most important, as being of frequent recurrence and reducible under general laws. 1st Form. Factorial expressions of the form 2 (x - 1) ... (o — m + 1) or a(m) in the notation of Ch. II. Art. 2. We have Axfm+1) = (m + 1)(m); 2(m+1) .. ) + C, M +1 (26 — 1)... (xor Ex(x - 1)...(ac — m + 1) = m) + C ...... (1). mm +1 Taking this between limits X = n and x =m, (n > m), we get 1.2...m+2.3... (m+1)+...+(n- m)... (n − 2) (n − 1) n(n-1)... (n - m) .m+1 B. F. D. 5 Or we may retain C and determine it subsequently, thus 1.2...m +2.3...(m +1)+...+(n- m)... (n − 2)(n-1) n(n − 1) ... (n – m) m +1 Put n=m+1 and the series on the left-hand side reduces to its first term, and we obtain (m + 1)m ... 1 1.2 + C; .C=0. m +1 + C. m = Here a = 2, b=5, m= 3, and since we have to find the sum of n terms we must change n into n+1 in the last formula, and we obtain (2n+7) (2n +5)(2n+3) 4 x 2 2nd Form. Factorial expressions of the form 1 + 1.6.0 We have by Ch. II. Art. 2, Ax(=m+1) = (-m + 1)2m); It will be observed that there must be at least two factors in the denominator of the expression to be integrated. No 1 finite expression exists for E ax +59 Ex. 2. Find the sum of n terms of the series 1 1 + + &c. 1.4.7 ' 4.7.10 We have here a= 3, b=-2, m = = 3. Un Unt Unt2 .: Sum of (n − 1) terms 1 1 = Σ. C 3 x 2 x Un • Unt1 1 EC 6 (3n— 2) (3n + 1) Put n=2 and we obtain 1 1 1 с .. C= 1.4.7 6.4.7 24 Hence (writing n for n-1 and therefore n + 1 for n) 1 1 Sum of n terms 24 6 (3n + 1)(3n+4) As all that is known of the integration of rational functions is virtually contained in the two primary theorems of (2) and (5), it is desirable to express these in the simplest form* Supposing then Ug = ax + b, let U ,Uz-, ... Uz_m+1 = (ax + b)(), = (ax + b)(m), (ax + b)(m+2) Σ (αα + β)»), + C ......... (6), a (m+1) whether m be positive or negative. The analogy of this result with the theorem (ax + b)*+1 s (ax +b)" dx +C a (m +1) is obvious. We shall now shew how to reduce other forms to one of the preceding. 3rd Form. Rational and integral functions. * As most of the summations of series whose nth term is a rational function of n will have to be effected by these methods, and as such summations are of very frequent occurrence, it is still more important to have a readily applicable rule for effecting them. The following is perhaps the most convenient form for finding the sum of n terms of such series:“Write down the nth term with its factors in ascending order of mag Sadd one factor at the end nitude, divide by the number of factors now remaining, and by the coefficient of X (in each factor), and Sadd to Tsubtract from a constant." It is scarcely necessary to add that the upper line in the brackets must be taken when the terms are of the form Uz Uz Uz-m+1 and the lower when 1 of the form u Uzt1 Ut+m=1 |